Problem 15 - Chemistry in 30 days

Question

Equal masses of methane and hydrogen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). The fraction of the total pressure exerted by hydrogen is: (a) \(1 / 2\) (b) \(8 / 9\) (c) \(1 / 9\) (d) \(16 / 17\)

Step-by-Step Solution

Answer

The fraction of the total pressure exerted by hydrogen is \( \frac{8}{9} \).

Step 1: Convert Mass to Moles

Calculate the number of moles of each gas using their molecular weights.The molecular weight of methane ( ext{CH}_4) is 16 g/mol, and that of hydrogen ( ext{H}_2) is 2 g/mol. Given equal masses, let the mass of each gas be m grams.For methane, moles = \( \frac{m}{16} \).For hydrogen, moles = \( \frac{m}{2} \).

Step 2: Calculate Total Moles

Add the moles of each gas to find the total moles of gas in the container.Total moles = \( \frac{m}{16} + \frac{m}{2} \).To simplify, find a common denominator:Total moles = \( \frac{m}{16} + \frac{8m}{16} = \frac{9m}{16} \).

Step 3: Determine Mole Fraction of Hydrogen

Calculate the mole fraction of hydrogen in the mixture.The mole fraction for hydrogen is:\[ \text{Mole fraction of hydrogen} = \frac{\frac{m}{2}}{\frac{9m}{16}} \]Simplifying further:\[ \frac{m}{2} \times \frac{16}{9m} = \frac{16}{18} = \frac{8}{9} \]

Step 4: Calculate Pressure Fraction

The pressure each gas exerts in a mixture of gases is proportional to its mole fraction.Thus, the fraction of the total pressure exerted by hydrogen is equal to its mole fraction:Fraction of total pressure by hydrogen = \( \frac{8}{9} \).

Key Concepts

Mole Fraction

The mole fraction is a way to express the concentration of one component in a mixture. It's simply the ratio of the number of moles of a particular substance to the total number of moles of all substances present. This can be thought of as a kind of 'share' each gas has in the total. For instance, if you have water vapor mixed with air, calculating the mole fraction tells you how much of the mixture is water. To compute this, you use the formula:

• Mole Fraction = \( \frac{\text{Moles of Component}}{\text{Total Moles}} \)

In our exercise, we calculated the mole fraction of hydrogen by dividing its moles by the total moles of the gases. This helps provide insight into how much pressure each gas will exert, as mole fraction directly relates to pressure exertion in a gas mixture. It's a crucial concept in understanding gas behavior in mixtures.

Molecular Weight

Molecular weight, also known as molecular mass, is the weight of all the atoms in a molecule. It's a critical value that allows us to convert between grams and moles. Each element on the periodic table has an atomic mass, and by adding these up for a molecule, you obtain its molecular weight.For example:

• Methane (\(\text{CH}_4\)) has a molecular weight of 16 g/mol.
• Hydrogen (\(\text{H}_2\)) has a molecular weight of 2 g/mol.

Understanding molecular weight is important when you're working with gases, especially when you want to find out how many moles are in a given mass. This information is key for calculations involving chemical reactions and gas laws, where moles are a standard unit of measure.

Pressure Exertion

In a gaseous mixture, each gas contributes to the total pressure in proportion to its mole fraction. This concept is derived from Dalton's Law of Partial Pressures, which states that the total pressure of a gas mixture is the sum of the pressures of each individual gas if it were alone in the same volume. Thus, if you know the mole fraction of a gas, you can directly figure out how much pressure it exerts:

• Pressure exerted by a gas = Mole Fraction of Gas × Total Pressure of Mixture

This was used in the exercise to find that the fraction of the total pressure exerted by hydrogen is equal to its mole fraction. This relationship simplifies the calculation of pressures when dealing with mixtures, especially in understanding and applying the gas laws.

Moles Calculation

Moles are a fundamental concept in chemistry representing a specific quantity of particles or molecules. When dealing with gases, calculating the number of moles present is essential. This value helps translate mass into a countable amount of molecules, which is pivotal for stoichiometric calculations and gas law applications.The formula to convert mass into moles involves molecular weight:

• Moles = \( \frac{\text{Mass}}{\text{Molecular Weight}} \)

In our exercise, equal masses of methane and hydrogen were compared. We calculated moles using their respective molecular weights. This step helped in determining each gas's contribution to the total moles and hence, the pressure exerted by each in the container. Having an accurate mole calculation is indispensable for understanding gas behaviors and interactions.